Theorem felapton 2159

Description: "Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
felapton.maj	⊢ ∀𝑥(𝜑 → ¬ 𝜓)
felapton.min	⊢ ∀𝑥(𝜑 → 𝜒)
felapton.e	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
felapton	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem felapton

Step	Hyp	Ref	Expression
1		felapton.e	. 2 ⊢ ∃𝑥𝜑
2		felapton.min	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜒)
3	2	spi 1550	. . 3 ⊢ (𝜑 → 𝜒)
4		felapton.maj	. . . 4 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
5	4	spi 1550	. . 3 ⊢ (𝜑 → ¬ 𝜓)
6	3, 5	jca 306	. 2 ⊢ (𝜑 → (𝜒 ∧ ¬ 𝜓))
7	1, 6	eximii 1616	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)